Set Theory and Indiscernibles Ali Enayat IPM Logic Conference June - - PDF document

Set Theory and Indiscernibles Ali Enayat IPM Logic Conference June 2007

LEIBNIZ’S PRINCIPLE OF IDENTITY OF INDISCERNIBLES

• The principle of identity of indiscernibles,

formulated by Leibniz (1686), states that no two distinct substances exactly resem- ble each other.

• Leibniz’s principle can be construed as pre-

scribing a logical relationship between ob- jects and properties: any two distinct ob- jects must differ in at least one property. This suggests a model theoretic interpre- tation:

• Fix a model M = (M, · · ·) in a language L,

let the “objects” refer to the elements of M, and the “properties” refer to properties that are L-expressible in M via first order formulas with one free variable.

LEIBNIZIAN MODELS

• Let us call a model M to be Leibnizian iff M

contains no pair of distinct elements a and b, such that for every first order formula ϕ(x) of L with precisely one free variable x,

M ϕ(a) ↔ ϕ(b).

• Any pointwise definable model is Leibnizian,

e.g., (ω, <), (Vω, ∈), and (L(ωCK

1

), ∈).

• Any model M = (M, · · ·) in a language L

such that |M| > 2|L|.ℵ0 is not Leibnizian.

• Every Leibnizian model is rigid, but not

vice versa: (ω1, <) is rigid but not Leib- nizian.

LEIBNIZIAN MODELS, CONT’D

• The field R of real numbers, and the ring
• f integers Z are both Leibnizian, but the

field C of complex numbers is not.

• Every Archimedean ordered field is Leib-

nizian.

• Moreover, Tarski’s elimination of quanti-

fiers theorem for real closed fields implies that the Leibnizian real closed fields are precisely the Archimedean real closed fields.

• Non-Archimedean Leibnizian ordered fields

exist in every infinite cardinality ≤ 2ℵ0.

THE LEIBNIZ-MYCIELSKI AXIOM (LM)

• Leibniz’s principle cannot be expressed in

first order logic, even for countable struc- tures. This is an immediate corollary of Ehrenfeucht-Mostowski’s theorem on indis- cernibles.

• However, Mycielski (1995) has introduced

the following first order axiom (LM) in the language of set theory {∈}which captures the spirit of Leibniz’s principle for models

• f set theory:

∀x∀y [x = y → ∃α > max{ρ(x), ρ(y)} Th(Vα, ∈, x) = Th(Vα, ∈, y)].

• Theorem (Mycielski). A complete exten-

sion T of ZF proves LM iff T has a Leib- nizian model.

LM AS A CHOICE PRINCIPLE

• Kinna-Wagner Selection Principles (1955)

KW1: For every family A of sets there is a function f such that ∀x ∈ A (|x| ≥ 2 → ∅ = f(x) x). KW2: Every set can be injected into the power set of some ordinal. ZF ⊢ KW1 ← → KW2.

• GKW1:

There is a definable (without pa- rameters) map F such that F(x) x) for every x with two or more elements.

• GKW2:

There is a definable (without pa- rameters) map G such that G injects V into the class of subsets of Ord”.

THE EQUIVALENCE OF LM WITH GLOBAL KM Theorem. Suppose M is a model of ZF. The following are equivalent: (i) M satisfies GKW1. (ii) M satisfies GKW2. (iii) M satisfies LM.

COROLLARIES Corollary. ZF + LM ⊢ KW.

• Corollary. ZF + V = OD ⊢ LM.

Corollary In the presence of ZF + LM there is a parameter free definable global linear or- dering of the universe. Corollary. ZF+LM proves GC<ω(global choice for collections of finite sets). Corollary. ZF + LM proves the existence

• f a definable set of real numbers that is not

Lebesgue measurable.

OPEN QUESTIONS

• Question 1.

(Abramson and Harrington, 1977). Does every completion T of ZF have an uncountable model without a pair

• f indiscernible ordinals?
• Question 2 (Schmerl).

Is there a model

• f set theory with a pair of indiscernibles,

but not with a triple of indiscernibles?

ANTI-LEIBNIZIAN SYSTEMS ZFC(I) is a theory in the language {∈, I(x)}, where I(x) is a unary predicate [but we shall write x ∈ I instead of I(x)], whose axioms are:

• ZFC + All instances of replacement in the

language {∈, I(x)};

• I is a cofinal subclass of ordinals:

(I ⊆ Ord)∧ ∀x ∈ Ord ∃y ∈ Ord (x ∈ y ∈ I);

• For each n-ary formula ϕ(v1, · · ·, vn) in the

language {∈}, ∀x1 < · · · < xn, ∀y1 < · · · < yn from I ϕ(x1, · · ·, xn) ↔ ϕ(y1, · · ·, yn).

ZFC(I) AND LARGE CARDINALS

• If κ is a Ramsey cardinal, then (Vκ, ∈) ex-

pands to a model of ZFC(I).

• If (Lκ, ∈) expands to a model of ZFC(I),

and cf(κ) > ω, then 0# exists.

• If 0# exists, then L cannot be expanded to

ZFC(I).

• Every well-founded model of ZFC(I) sat-

isfies 0# exists.

THE SYSTEM ZFC(I<ω)

• ZFC(I<ω) is a theory in the language

{∈} ∪ {In(x) : n ∈ ω}, where each In is a unary predicate, whose axioms are:

• ZFC + All instances of replacement in the

language {∈} ∪ {In(x) : n ∈ ω};

• In is a cofinal subclass of ordinals;
• I0 is a class of indiscernibles for (V, ∈), and

for n ≥ 0, In+1 is a class of indiscernibles for the structure (V, ∈, I0, · · ·, In).

• Question. What are the consequences of

ZFC(I) and ZFC(I<ω) in the ∈-language

• f set theory?

THE ANSWER

• Theorem.

The following are equivalent for a completion T of ZFC:

• 1. Some model of T expands to a model of

ZFC(I).

• 2. Some model of T expands to a model of

ZFC(I<ω).

• 3. Some model of T expands to a model of

GBC + “Ord is weakly compact”.

• 4. T is a completion of ZFC + Φ.
• GBC = G¨
• del-Bernays class theory.
• “Ord is weakly compact” is the statement

“every Ord-tree has a branch”.

THE CANONICAL SET THEORY ZFC + Φ

• Φ := {∃θ (θ is n-Mahlo and Vθ ≺n V) :

n ∈ ω}

• Φ0 := { ∃θ (θ is n-Mahlo) : n ∈ ω}.
• Over ZF, Φ and Φ0 are equivalent.
• Motto: Φ allows infinite set theory to catch

up with finite set theory, vis-` a-vis Model Theory.

A KEY EQUIVALENCE

• Theorem.
• 1. If (M, A) GBC + “Ord is weakly com-

pact”, then M ZFC + Φ.

• 2. Every consistent completion of ZFC + Φ

has a countable model which has an expan- sion to a model of GBC + “Ord is weakly compact”. [Schmerl-Shelah (1972) Kaufmann (1983) E(1987, 2004)]

CONCLUDING CONSIDERATIONS

• If Replacement(I) is weakened to Separation(I)

in ZFC(I), while retaining “I is cofinal”, then the resulting theory is conservative

• ver ZFC.
• We can strengthen ZFC(I) to ZFC(I+)

with “C is a cub” to ensure that when the indiscernibles are stretched, a model with a least new ordinal is obtained.

• ZFC(I+) turns out to be a conservative

extension of a ZFC +Ψ, where the scheme Ψ is obtained from Φ by replacing “n-Mahlo” by “n-subtle”, i.e., the axioms of Ψ are of the form “∃θ(θ is n-subtle and Vθ ≺n V”.